I recently came back from holiday in Chiang Mai, and I was consistently surprised by how young Thailand is, how the cities don't seem to be very old (compared to, say, China). I went to a few museums that seemed to indicate the Thais migrated to what is now Thailand only about 1000 years ago, but other things seemed older, so it was rather confusing.
I want to change your question a little bit and instead of speaking about Thais, to speak instead about Tais, since that's a more appropriate focus for the question. The Tais, of whom the Thai people in Thailand are just a part, are incredibly widespread throughout SE Asia. You find them in Assam (Tai Ahom (for whom the word Assam may possible be taken), Tai Khamti, Tai Phake), in Burma (Tai Shan (which may be etymologically related to that same "Assam" name) among others), in Laos (the language is a Tai language), in Vietnam, and of course still in China (where they're called Dai 傣 but it's pronounced the same as Tai).
There are other groups as well, such as Zhuang, but excluding them, the rest of the groups I've mentioned are much more similar and grouped together linguistically more closely to each other than to Zhuang. The point being, the Tais are a geographically massive group.
Each of these groups tie (heh) back to a group in SW China which, over multiple migrations, spread throughout the greater part of mainland SE Asia, and in many cases have remained in the various places they stopped on the way. The Tais are easily one of the most widepsread groups throughout SE Asia.
The Ahoms migrated to the area of NE India out of the Yunnan/Burma border area around the 13th century. Their king, Sukaphaa, is still highly regarded both by Ahoms and Indic Assamese today. At its founding, what I'll call Assam began as a Mueang^1, a sort of semi-independent city state found throughout the Tai/Thai sphere of influence. These were spread around the region covering most of SE Asia as well as parts of China and Assam as a distinctly Tai political structure. The muangs in their most typical form were independent walled cities with single rulers, but functioned in an extended tribute network with each other and more distant powers.
Mong Mao, the Tai kingdom from which Sukaphaa and his people migrated, had existed in the area of Northern Burma since around the 10th century, and the area is still inhabited primarily by Tais today.
It is around that time, or possibly a couple centuries before, that the Tais/Thais first migrated out of SW China and into what is today Burma. Like with all major southward migrations, a.k.a. the entire history of historical migrations in E/SE Asia, population pressures and successive waves of northern migrants forced the Tais south. The same pressures will come up again and again throughout the millennia in Greater China, and for just about every group the "why did they move" question has this as its answer.
I went to a few museums that seemed to indicate the Thais migrated to what is now Thailand only about 1000 years ago, but other things seemed older
This is accurate, on both counts. The Thais arrived in the Chao Phraya river valley (the central part of what's today Thailand) around the 10th century, but they didn't arrive in an unpopulated jungle. There were already people living i the area. The Khmers were already there at least two centuries earlier. The Mon were as well as Hariphunchai in what is now Northern Thailand. In addition to the Mon and the Khmer, a number of other groups were also there before the arrival of the Thais. A much earlier kindgom exists as well, known as Funan in Chinese, but the details aren't so well known.
If you check out the Wikipedia page on Tai-Kadai languages, you'll see a map where orange represents SW Tai languages. Actually that's less area than currently inhabited by Thais/Tais in SE Asia. For example upper Assam has a huge Tai population, but the Tai Ahoms have lost their spoken language^2 and are now speaking Assamese, an Indo-Aryan language close to Bengali.
Anyway I hope that helps answer your question. Let me know if I can clear anything up.
tl;dr:
When: ~10th century CE
Why: Population pressures from the north
See A History of Thailand by Baker & Phongpaichit for a good overview of the Mueang system.
They still have a lot of manuscripts though, which are super fascinating. Tai isn't one of my main research focuses but in the last year or so I've worked with a number of Tai communities in different parts of SE Asia and it's been interesting to see the very clear similarities in language and culture, despite a long history over a huge area.